Ricocheting inclined layer convection states

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Ricocheting inclined layer convection states

Laurette Tuckerman

To cite this version:

Laurette Tuckerman. Ricocheting inclined layer convection states. Journal of Fluid Mechanics,

Cam-bridge University Press (CUP), 2020, 900, pp.F1-F4. �10.1017/jfm.2020.575�. �hal-02995000�

Focus on Fluids 1

Ricocheting inclined

layer convection states

Laurette S. Tuckerman

1

1

Physique et M´ecanique des Milieux H´et´erog`enes (PMMH), ESPCI Paris, CNRS, PSL Research University, Sorbonne Universit´e,

Universit´e de Paris, F-75005 Paris, France (Received 5 January 2021)

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Inclining a fluid layer subjected to a temperature gradient introduces a profusion of fascinating patterns and regimes. Previous experimental and computational studies form the starting point for an extensive numerical bifurcation study by Reetz & Schneider (2020a); Reetz et al. (2020b). Intricate trajectories passing through multiple steady and periodic states organize the dynamics. The consequences for chaotic patterns in large geometries is discussed.

Keywords. B´enard convection, pattern formation, bifurcations.

1. Introduction: inclined layer convection

The phenomenon of thermal convection is known to all: warm fluid rises and cool fluid falls. But the fluid cannot rise and fall in the same locations; the way in which the rising and falling locations are arranged in space provides the archetypal example of pattern formation. Reetz and coworkers (2020a; 2020b) explore a generalization of Rayleigh-Benard convection in which the confining plates held at different temperatures are inclined at an angle γ with respect to gravity. From a physical point of view, the immediate consequence is a shear flow in the direction of inclination, upwards along the warmer plate and downwards along the cooler one. From a mathematical point of view, the inclination renders the fluid layer anisotropic, distinguishing the direction of inclination and that perpendicular to it. As the Rayleigh number is increased, inclined convection undergoes a first transition from a featureless base state B to straight rolls. For compressed CO2 with a Prandtl number of 1.07, these rolls are buoyancy-driven,

with axes parallel to the inclination (longitudinal rolls, LR seen in the title figure) for γ . 78◦_{, and shear-driven, with axes perpendicular to the inclination (transverse rolls,}

T R seen in the title figure) for γ & 78◦ _{(e.g. Gershuni & Zhukhovitskii 1969; Chen &}

Pearlstein 1989), as shown in Figure 1. The limiting case in which the bounding plates are vertical (γ = 90◦) and the imposed temperature gradient is horizontal is sometimes called natural convection.

The problem of inclined layer convection received a boost from experimental obser-vations of exotic regimes such as crawling rolls and transverse bursts by Daniels and co-workers (Daniels et al. 2000; Daniels & Bodenschatz 2002; Daniels et al. 2003). These were followed by numerical and theoretical simulations of these patterns by timestepping (Daniels et al. 2008) and by Floquet and Galerkin analyses (Subramanian et al. 2016).

2 Laurette S. Tuckerman

15

0 500 1000 1500 2000 2500 3000 t 0.7 0.8 0.9 1.0 1.1 I LR W R 0.95 1.00 1.05 D/I 0.7 0.8 0.9 1.0 1.1 I B LR OW R OW R W R (a) (b) (c)

Figure 6: State trajectory evolution from the unstable base state B at

= 40 and

Ra = 3344 (✏ = 0.5). The trajectory visits unstable LR, followed by a transient visit of

W R (inset (c)). Subsequently, the trajectory undergoes a sequence of rapid oscillations

and is finally attracted to a heteroclinic cycle between equilibrium state OW R and

a symmetry related equilibrium. The time series for 2000 < t < 10 000 in inset (a)

indicates the increasing time spent near the equilibrium states. The phase portrait of

the heteroclinic cycle is magnified in inset (b). Since the energy transfer rates do not

di↵er for symmetry related states, the heteroclinic cycle appears as a homoclinic cycle.

See Figure 7a for a projection that distinguishes the two equilibrium states.

equilibrium state ⌧

xy

OW R (Figure 7). Here, ⌧

xy

= ⌧ (0.25, 0.25) is a shift operator

trans-lating the wavy flow structure by half a pattern wavelength in the direction of the domain

diagonal along which the wavy convection rolls of OW R are aligned. OW R and ⌧

xy

OW R

show two spatial periods of the wavy pattern along the domain diagonal and both are

invariant under transformations of S

owr

=

h⇡

xyz

, ⌧ (0.5, 0.5)

i. Without imposing the

sym-metries in S

owr

, OW R and ⌧

xy

OW R have each a single purely real unstable eigenmode,

denoted as state vector e

u

_{and ⌧}

xy

e

u

, respectively, that breaks the ⌧ (0.5, 0.5)-symmetry

by having only one spatial period along the domain diagonal (Figure 7c). Perturbing

OW R with e

u

_{initiates the temporal transition OW R}

_{! ⌧}

xy

OW R. Perturbing ⌧

xy

OW R

with ⌧

xy

e

u

initiates the temporal transition ⌧

xy

OW R

! OW R. Together, the two

dy-namical connections form a heteroclinic cycle. The two involved unstable eigenmodes

e

u

_{and ⌧}

xy

e

u

preserve the symmetries ⇡

xyz

and ⇡

xyz

⌧ (0.5, 0.5), respectively. Thus, they

are analogous to sine- and cosine-eigenmodes in that they first, are orthogonal such

that the L

2

inner product is

he

u

, ⌧

xy

e

u

i = 0 and second, have a reflection symmetry

with respect to di↵erent reflection points, namely ⇡

xyz

and ⇡

xyz

⌧ (0.5, 0.5) (Figure 7c).

When imposing ⇡

xyz

-symmetry, the unstable eigenmode ⌧

xy

e

u

is disallowed, ⌧

xy

OW R

becomes dynamically stable, and the associated symmetry subspace ⌃

1

contains only the

dynamical connection OW R

_{! ⌧}

xy

OW R. When imposing ⇡

xyz

⌧ (0.5, 0.5)-symmetry, the

unstable eigenmode e

u

_{is disallowed, OW R becomes dynamically stable, and the}

associ-ated symmetry subspace ⌃

⌧

contains only the dynamical connection ⌧

xy

OW R

! OW R.

Hence, the heteroclinic cycle satisfies all three conditions for the existence of a robust

heteroclinic cycle between two symmetry related equilibrium states (Krupa 1997):

(i) OW R is a saddle and ⌧

xy

OW R is an attractor (or sink) in a symmetry subspace

⌃

1

of the entire state space of the [2

x

,

y

]-periodic domain.

(ii) There is a saddle-attractor connection OW R

_{! ⌧}

xy

OW R in ⌃

1

.

Page 15 of 30

Figure 1. Left: evolution from base state B to longitudinal rolls LR to wavy rolls W R (inset (c)), followed by a sequence of widening oscillations leading to a heteroclinic cycle between the oblique wavy roll state OW R and its image under a shift τxyin x and y (inset (b)). The trajectory

is projected onto coordinates (D/I, I), with D the viscous dissipation and I the energy input. Right: temperature isosurfaces of W R and OW R. Adapted from Reetz & Schneider (2020a).

2. Overview: a numerical bifurcation study

Reetz and coworkers (2020a; 2020b) have sought the bifurcation-theoretic origin of these exotic patterns. Their motivation is in part because of the features that inclined layer connection shares with wall-bounded shear flows such as plane Couette flow and pipe flow. These shear flows were not known to have any non-trivial solutions until their resistance to bifurcation-theoretic approaches was breached by Nagata (1990) and Clever & Busse (1992), who continued solutions from Taylor-Couette flow and Rayleigh-B´enard convection to plane Couette flow. Shortly thereafter, traveling waves in pipe flow were computed by Faisst & Eckhardt (2003) and Wedin & Kerswell (2004). Since then, dozens of solutions have been computed (e.g., Gibson et al. 2008), all unstable.

These multiple solutions are thought to be of more than zoological interest, since Cvitanovi´c & Eckhardt (1991); Kawahara et al. (2012) have proposed that turbulence in shear flows could be viewed as a collection of trajectories ricocheting between these multiple solutions via the connections between them. Indeed, for pipe flow, Hof et al. (2004) compared exact solutions with snapshots during the evolution of the flow.

Reetz & Schneider (2020a) focus on several key states and dynamical regimes. One example is a sequence of bifurcations at γ = 40◦ from the base state to longitudinal rolls to wavy rolls to oblique wavy rolls. The oblique wavy rolls participate in a robust heteroclinic cycle, shown in figure 2. Reetz et al. (2020b) expand on these results by surveying the parameter space of Rayleigh number and inclination angle to construct complete bifurcation diagrams and interpreting the transitions in the context of the large-aspect-ratio experiments. They have computed eight different invariant solutions all at Rayleigh number 21 266 and angle γ = 90◦_{, where the longitudinal rolls (LR) do not}

exist: the transverse rolls (TR) already mentioned, and various periodically modulated versions of the longitudinal rolls. Carrying out direct numerical simulations at the same parameter values, they extracted snapshots that resembled these states, shown in figure 2.

Ricocheting inclined layer convection states 3

Figure 2. (a)-(h) Eight invariant states in vertical layer convection (γ = 90◦), all at Rayleigh number 21 266, computed by continuation. (i)-(l) Four snapshots from DNS of turbulent regime at the same parameters. Adapted from Reetz et al. (2020b).

3. Future: from bifurcation diagrams to turbulence?

There is no doubt that patterns and temporal behavior are controlled and explained by the plethora of underlying dynamical objects – fixed points and traveling waves, pe-riodic orbits and heteroclinic cycles. The ability to compute such objects for the full three-dimensional Navier-Stokes and Boussinesq equations has resulted from advances in several fields: first, the spectacular growth of dynamical systems theory and symmetry (e.g., the monographs by Golubitsky et al. 1988; Kuznetsov 1998) following the discovery of deterministic chaos; second, the increasing power of computation; and third, the incor-poration of matrix-free methods for linear algebra (Dijkstra et al. 2014) into algorithms for tracking dynamical objects. See, e.g., Marques et al. (2007); Boro´nska & Tuckerman (2010a,b) for computational bifurcation studies of convection. However, the application of the theories of Cvitanovi´c & Eckhardt (1991); Kawahara et al. (2012) to a turbulent hydrodynamic state would require the computation of an even larger number of dynami-cal objects, on a sdynami-cale that is not yet possible. The challenge is to bridge the gap between bifurcation theory and the large-scale statistical phenomenon of turbulence.

Conflicts of interest

The author reports no conflict of interest.

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4 Laurette S. Tuckerman

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